In a recent series of papers [1-3], we have discovered a connection between non-perturbative quantum field theory and the (noncommutative) geometry of configuration spaces. In this blog-post, we would like to outline our findings in a non-technical manner intended for readers, who are familiar with the basics of noncommutative geometry and high-energy physics.
The noncommutative standard model
Chamseddine and Connes and co-workers have shown that the standard model of particle physics coupled to general relativity can be formulated in terms of an almost-commutative spectral triple [4-8]. Their seminal work, which renders key aspects of the standard model in a completely new light, raises two fundamental questions:
- where does the almost-commutative algebra, that underlies Chamseddine’s and Connes’ work, originate from? Is there a deep reason why Nature chose this algebraic structure?
- what role should quantum field theory play in this approach to fundamental physics? As it stands now, perturbative quantum field theory is applied to Chamseddine’s and Connes’ construction as something like an afterthought and only to the standard model part: gravity is not quantized.
Embedded within this last question lies also the question of whether gravity should be quantized.
In our recent work, we propose a novel answer to these two questions. It turns out that a geometrical construction over a configuration space of gauge connections gives rise to a non-perturbative quantum field theory on a curved background — including both bosonic and fermionic degrees of freedom — which produces an almost-commutative algebra similar to that of Chamseddine and Connes. Thus, what we propose is that the standard model of particle physics originates from a geometrical framework intimately related to non-perturbative quantum field theory.
For another possible answer to the first question see also the work of Connes, Mukhanov, and Chamseddine [9,10].
The geometry of moving stuff around
The starting point of our construction is an algebra called the HD algebra [11,12], which is generated by parallel transports along flows of vector fields in a three-dimensional manifold. That is, the HD algebra encodes how tensor-degrees of freedom are moved around in space. The HD algebra, which comes with a high degree of canonicity, is a non-commutative algebra of functions over a configuration space of gauge connections.
The next step is to formulate an infinite-dimensional Bott-Dirac operator on the configuration space of gauge connections. The construction of this operator, which resembles an infinite-dimensional Bott-Dirac operator that Kasparov and Higson constructed in 2001 , is severely restricted by the requirement of gauge-covariance.
A key feature of the Bott-Dirac operator is that its square produces the Hamilton operator of a Yang-Mills theory coupled to a fermionic sector as well as a topological Yang-Mills term together with higher-order terms — all on a curved background. Furthermore, the Bott-Dirac operator and its interaction with the HD algebra reproduce the canonical commutation relations of a quantized gauge and fermionic fields.
Thus, the formulation of a noncommutative geometry on a configuration space of gauge connections gives rise to the basic building blocks of a non-perturbative Yang-Mills-Dirac theory on a curved background.
The origin of fermionic quantum field theory
One interesting feature of this construction is the role fermions play. The Bott-Dirac operator requires an infinite-dimensional Clifford algebra — just as an ordinary Dirac operator requires a finite-dimensional Clifford algebra — and it is this Clifford algebra that gives us the CAR algebra and the fermionic Fock space. Furthermore, when we compute the square of the Bott-Dirac operator then a certain commutator turns up, which gives us precisely the Dirac Hamilton operator. Thus, the Dirac Hamiltonian can be understood as a quantum fluctuation of the bosonic theory.
All this shows that the fermionic degrees of freedom play an intrinsically geometrical role in this framework; they are intimately related to the geometry of the underlying configuration space.
A link to the standard model
In a semiclassical limit, the HD-algebra will give rise to a matrix algebra. If we choose a configuration space of spin connections then the HD-algebra will give us a three-by-three matrix algebra in a classical limit. The construction of the Bott-Dirac operator gives, however, rise to additional structure, which means that in a classical limit we find an almost-commutative algebra with a matrix factor given by:
M_3(C) + M_3(C) + M_2(C)
which looks surprisingly similar to the matrix factor, which Chamseddine and Connes have found in the standard model [4,5]:
C + H + M_3(C)
where H is the quaternions. Could there be a connection here? More analysis is required to determine whether this link can be substantiated, but for now, we are encouraged by our findings.
Dynamical gauge-covariant regularisation
A key question is whether a Hilbert space representation of the HD algebra and the Bott-Dirac operator exists. We have shown that such a representation (strongly continuous, separable) exists in the special case where the gauge-symmetry is broken , but we do not have a proof in the gauge-covariant case. We are, however, confident that such representations do exist.
A key feature of the representation, which we have found, is a UV-regularisation in the form of a Sobolev norm that dampens degrees of freedom beyond a certain scale. In order to obtain a gauge-covariant representation, it is natural to make this Sobolev norm gauge-covariant. This is possible but in doing so we change the entire construction.
Why is that? Well, normally a UV-regularisation is a computational artifact, that should ultimately be removed. This is the origin of renormalization theory. There are of course several reasons why a regularisation must be removed, one being that there are countless ways of regularising and no way to choose between them. But this is no longer the case with a gauge-covariant regularisation because it will have a time-evolution, i.e. it will be dynamical. This is what we call a dynamical gauge-covariant regularisation .
What we find is that the UV-regularisation should be understood not as a computational artifact but as a physical feature: the UV-regularisation is part of the metric data of the configuration space. One thing that makes this possible is that from a perturbative QFT perspective this UV-regularisation will simply look like higher-order derivative terms. The non-locality, that the dynamical regularisation introduces, will only be seen non-perturbatively.
Note that the concept of a dynamical regularisation is almost inevitable in a non-perturbative gauge theory since any Hilbert space representation will likely require some kind of UV-regularisation. The requirement that this regularisation is gauge-covariant automatically makes it dynamical.
It is widely believed that a theory of quantum gravity will give rise to a Planck-scale screening, which in turn will impact the Lorentz symmetry and the causal structure of space-time. If, however, the Planck-scale screening originates from a framework of non-perturbative quantum field theory, as we suggest, then it would make a theory of quantum gravity obsolete. The regime, where such a theory would otherwise reign, would be screened off. We find this idea intriguing: perhaps the reason why a theory of quantum gravity has eluded theoretical physicists for so long is that it does not exist?
We believe that the idea of a dynamical UV-regularisation deserves much attention. After all, there exist very few concrete ideas as to how a Planck-scale screening may arise in Nature. This is one such idea.
In our recent work, we have shown that essentially all the key building blocks of modern high-energy physics — bosonic and fermionic quantum gauge theory, gravity, an almost-commutative algebraic structure — emerges from a simple geometrical framework on a configuration space. This work raises many questions, some of the most important ones are:
- does a gauge-covariant Hilbert space representation exist?
- what does the construction look like when we use the Levi-Civita connection on the configuration space (we know that the Levi-Civita connection exists).
- what does the time-evolution of the UV-regularisation look like?
- we assume that the non-locality will be restricted to scales beyond the Planck scale. What happens to causality? The Lorentz symmetry?
- can the conjectured connection to the standard model be substantiated? In particular, does it offer an answer to the question of why there are three particle generations?
With this, we end this blog-post;
 J. Aastrup and J. M. Grimstrup, “The Metric Nature of Matter” arXiv: 2008.09356.
 J. Aastrup and J. M. Grimstrup, “Non-perturbative Quantum Field Theory and the Geometry of Functional Spaces," arXiv:1910.01841.
 J. Aastrup and J. M. Grimstrup, “Representations of the Quantum Holonomy-Diffeomorphism Algebra," arXiv:1709.02943.
 A. Connes, “Gravity coupled with matter and the foundation of noncommutative geometry," Commun. Math. Phys. 182 (1996) 155.
 A. H. Chamseddine and A. Connes, “Universal formula for noncommutative geometry actions: Unification of gravity and the standard model," Phys. Rev. Lett. 77 (1996) 4868.
 A. Connes, ”Noncommutative geometry and the standard model with neutrino mixing,'' JHEP 11 (2006), 081.
 A. H. Chamseddine, A. Connes and M. Marcolli, ”Gravity and the standard model with neutrino mixing,’' Adv. Theor. Math. Phys. 11 (2007) no.6, 991-1089.
 A. H. Chamseddine and A. Connes, “Why the Standard Model,’' J. Geom. Phys. 58 (2008), 38-47.
 A. H. Chamseddine, A. Connes and V. Mukhanov, “Geometry and the Quantum: Basics,’' JHEP 12 (2014), 098.
 A. H. Chamseddine, A. Connes and V. Mukhanov, “Quanta of Geometry: Noncommutative Aspects,’' Phys. Rev. Lett. 114 (2015) no.9, 091302.
 J. Aastrup and J. M. Grimstrup, “C*-algebras of Holonomy- Diffeomorphisms and Quantum Gravity II”, J. Geom. Phys. 99 (2016) 10.
 J. Aastrup and J. M. Grimstrup, “The quantum holonomy-diffeomorphism algebra and quantum gravity,’' Int. J. Mod. Phys. A 31 (2016) no.10, 1650048.
 N. Higson and G. Kasparov, "E-theory and KK-theory for groups which act properly and isometrically on Hilbert space", Inventiones Mathematicae, vol. 144, issue 1, pp. 23-74.